1: It should. No such nag-screens here with AIS isntalled… You might want to try a clean install or repair of either or both the extension and/or the internet security app.
2: No, you do not ‘need’ it. It offers additional privacy and site reputation functions over the web shield in Internet Security which focusses mainly/solely on blocking viruses. The browser plugin is optional, it’s function seems somewhat comparable to the privacy functions an extension like Adblock Plus offers.